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A topos topos

A quantum operation/quantum channel chan: * *chan \,\colon\, \mathscr{H} \otimes \mathscr{H}^\ast \to \mathscr{H} \otimes \mathscr{H}^\ast is called a noisy operation [Horodecki, Horodecki & Oppenheim 2003] or a unistochastic channel [Życzkowski & Bengtsson 2004] if it has an environmental representationhannel#QuantumChannelsAndDecoherence) where the environment/bath system \mathscr{B} is in its maximally mixed state:

chan(ρ)=1dim()tr (U(ρid )U ). chan(\rho) \;\;=\;\; \frac{1}{dim(\mathscr{B})} \mathrm{tr}^{\mathscr{B}} \Big( U \big( \rho \otimes id_{\mathscr{B}} \big) U^\dagger \Big) \,.


The theorem (herechannel#QuantumChannelsAsPartialTracesOfUnitariesOnTensors)) that every endo-quantum channel has an environmental representation is often advertized with the addendum that “… and such the environment may be chose to be in a pure state”, but in fact this is assumption is what the existing proofs rely on. For environments in non-pure states it is not clear that they can environmentally represent all quantum channels, and for noisy/unistochastic channels it is not to be expected that they exhaust all quantum channels.

But Müller-Hermes & Perry 2019 show that all unital quantum channels on qbits can be realized as noisy/unistochastic channels (with a bath of size at least 4).


Precursor discussion of the concept is due to:

The terminology “unistochastic channels” was introduced in:

Proof that all unital quantum channels on qubits are unistochastic (noisy operations) for a bath of size at least 4:

In one direction, assume that

chan: * * chan \,\colon\, \mathscr{H} \otimes \mathscr{H}^\ast \longrightarrow \mathscr{H} \otimes \mathscr{H}^\ast

is a completely positive map. Then by operator-sum decomposition there exists a set (finite, under our assumptions) inhabited by at least one element

s ini:S, s_{ini} \,\colon\, S \,,

and an SS-indexed set of linear operators

s:SE s:,withsE s E s=Id, s \,\colon\, S \;\;\; \vdash \;\;\; E_s \;\colon\; \mathscr{H} \longrightarrow \mathscr{H} \,,\;\;\;\; \text{with} \;\;\;\; \underset{s}{\sum} E_s^\dagger \cdot E_s \,=\, Id \mathrlap{\,,}

such that

chan(ρ)=sE sρE s . chan(\rho) \;=\; \underset{s}{\sum} \, E_s \cdot \rho \cdot E_s^\dagger \,.

Now take

S \mathscr{B} \,\equiv\, \underset{S}{\oplus} \mathbb{C}

with its canonical Hermitian inner product-structure with orthonormal linear basis (|s) s:S\big(\left\vert s \right\rangle\big)_{s \colon S} and consider the linear map

V: |ψ sE s|ψ|s. \array{ \mathllap{ V \;\colon\;\; } \mathscr{H} &\longrightarrow& \mathscr{H} \otimes \mathscr{B} \\ \left\vert \psi \right\rangle &\mapsto& \underset{s}{\sum} \, E_s \left\vert \psi \right\rangle \otimes \left\vert s \right\rangle \mathrlap{\,.} }

Observe that this is a linear isometry

ψ|V V|ψ =s,sψ|E s E s|ψs|sδ s s =ψ|sE s E sId|ψ =ψ|ψ. \begin{array}{ll} \left\langle \psi \right\vert V^\dagger V \left\vert \psi \right\rangle \\ \;=\; \underset{s,s'}{\sum} \left\langle \psi \right\vert E_{s'}^\dagger E_s \left\vert \psi \right\rangle \underset{ \delta_s^{s'} }{ \underbrace{ \left\langle s' \vert s \right\rangle } } \\ \;=\; \left\langle \psi \right\vert \underset{ Id }{ \underbrace{ \underset{s}{\sum} E_{s}^\dagger E_s } } \left\vert \psi \right\rangle \\ \;=\; \left\langle \psi \vert \psi \right\rangle \mathrlap{\,.} \end{array}

This implies that VV is injective so that we have a direct sum-decomposition of its codomain into its image and its cokernel orthogonal complement, which is unitarily isomorphic to dim()1dim(\mathscr{B})-1 summands of \mathscr{H} that we may identify as follows:

V()((|s 0)). \mathscr{H} \otimes \mathscr{B} \;\simeq\; V\big( \mathscr{H} \big) \oplus \Big( \mathscr{H} \otimes \big( \mathscr{B} \ominus \mathbb{C}\left\vert s_0 \right\rangle \big) \Big) \,.

In total this yields a unitary operator

U:((|s ini))V()((|s ini)) U \;\colon\; \mathscr{H} \otimes \mathscr{B} \,\simeq\, \mathscr{H} \oplus \Big( \mathscr{H} \otimes \big( \mathscr{B} \ominus \mathbb{C}\left\vert s_{ini} \right\rangle \big) \Big) \underoverset{}{}{\longrightarrow} V\big( \mathscr{H} \big) \oplus \Big( \mathscr{H} \otimes \big( \mathscr{B} \ominus \mathbb{C}\left\vert s_{ini} \right\rangle \big) \Big) \;\simeq\; \mathscr{H} \otimes \mathscr{B}

and we claim that this has the desired action if we couple the system to the pure bath state: |s 0\left\vert s_0 \right\rangle:

trace (U(|s iniρs ini|)U ) =s,strace (|sE sρE s s|) =s,ss|sδ s sE sρE s =sE sρE s =chan(ρ). \begin{array}{l} trace^{\mathscr{B}} \Big( U \big( \left\vert s_{ini} \right\rangle \rho \left\langle s_{ini} \right\vert \big) U^\dagger \Big) \\ \;=\; \underset{s,s'}{\sum} trace^{\mathscr{B}} \big( \left\vert s \right\rangle E_s \cdot \rho \cdot E_{s'}^\dagger \left\langle s' \right\vert \big) \\ \;=\; \underset{s,s'}{\sum} \underset{ \delta_{s}^{s'} }{ \underbrace{ \left\langle s' \vert s \right\rangle } } E_s \cdot \rho \cdot E_{s'}^\dagger \\ \;=\; \underset{s}{\sum} E_s \cdot \rho \cdot E_{s}^\dagger \\ \;=\; chan(\rho) \,. \end{array}



A linear isometry is a linear map

ϕ: 1 2 \phi \,\colon\, \mathscr{H}_1 \longrightarrow \mathscr{H}_2

between vector spaces equipped with (Hermitian) inner products | i{\langle \cdot \vert \cdot \rangle}_{i} (often: Hilbert spaces) which preserves these inner products, in that

ϕ()|ϕ() 2=| 2 \left\langle \phi(-) \vert \phi(-) \right\rangle_2 \;=\; \left\langle - \vert - \right\rangle_2

or equivalently in terms of adjoint operators:

ϕ ϕ=Id 1. \phi^\dagger \cdot \phi \,=\, Id_{\mathscr{H}_1} \,.

If also the reverse condition ϕϕ \phi \cdot \phi^\dagger holds, then ϕ\phi is called a unitary operator, which is the case iff ϕ\phi is surjective map.


Environmental representation of measurement channels

By the general theorem about environmental representations of quantum channels, every quantum measurement channel on a quantum system \mathscr{H} may be decomposed as

  1. coupling of \mathscr{H} to an environment/bath system \mathscr{B},

  2. unitary evolution of the composite system \mathscr{H} \otimes \mathscr{B},

  3. averaging the result over the environment states.

The way this works specifically for quantum measurement channels has precursor discussion von Neumann 1932 §VI.3 and received much attention in discussion of quantum decoherence following Zurek 1981 and Joos & Zeh 1985.

(independently and apparently unkowingly of the general discussion of environmental representations in Lindblad 1975)


(we shall restrict attention to finite-dimensional Hilbert spaces not to get distracted by technicalities that are irrelevant to the point we are after)

if |b init:\left\vert b_{\mathrm{init}} \right\rangle \,\colon\, \mathscr{B} denotes the initial state of a “device” quantum system then any notion of this device measuring the given quantum system \mathscr{H} (in its measurement basis WW, W\mathscr{H} \simeq \underset{W}{\oplus}\mathbb{C}) under their joint unitary quantum evolution should be reflected in a unitary operator under which [Zurek 1981 (1.1), Joos & Zeh 1985 (1.1.), following von Neumann 1932 §VI.3, review includes Schlosshauer 2007 (2.51)]:

  1. the system \mathscr{H} remains invariant if it is purely in any eigenstate |w\left\vert w \right\rangle of the measurement basis,

  2. while in this case the measuring system evolves to a corresponding “pointer state” |b w\left\vert b_w \right\rangle:

(1) unitary measurement interaction U W: |w,b ini |w,b w \array{ &\mathclap{ \color{green} \array{ \text{unitary} \\ \text{measurement interaction} } }& \\ \mathllap{ U_W \;\colon\;\; } \mathscr{H} \otimes \mathscr{B} &\longrightarrow& \mathscr{H} \otimes \mathscr{B} \\ \left\vert w, b_{ini} \right\rangle &\mapsto& \left\vert w, b_w \right\rangle }

for b inib_{\mathrm{ini}} and b wb_w distinct elements of an (in practice: approximately-)orthonormal basis for \mathscr{B}. (There is always a unitary operator with this mapping property (1), for instance the one which moreover maps |w,b w|w,b ini\left\vert w, b_{w}\right\rangle \mapsto \left\vert w, b_{\mathrm{ini}}\right\rangle and is the identity on all remaining basis elements.)

But then the composition of the corresponding unitary quantum channel with the averaging channel over \mathscr{B} is indeed equal to the WW-measurement quantum channel on \mathscr{H} (cf. eg. Schlosshauer 2007 (2.117), going back to Zeh 1970 (7)), as follows:

Last revised on September 28, 2023 at 06:54:18. See the history of this page for a list of all contributions to it.